An improved generalized normal distribution optimization and its applications in numerical problems and engineering design problems

Abstract

Generalized normal distribution optimization (GNDO) inspired by the theory of normal distribution is a recently developed metaheuristic method for global optimization problems. This work presents a novel variant of GNDO, which is called elite-driven generalized normal distribution optimization (EDGNDO). EDGNDO enhances the global search ability of GNDO by the designed search mechanism consisting of three local search operators and three global search operators that are based on two built archives used to save elite individuals. Note that, EDGNDO only needs population size and termination criteria for optimization, which can distinguish it over the most reported metaheuristic methods. The performance of EDGNDO is investigated by the well-known CEC 2017 test suite including three unimodal functions and 27 multimodal functions. Experimental results demenstrate that EDGNDO is obviously better than GNDO and the other five powerful algorithms in terms of solution quality and computational efficiency. In addition, EDGNDO is also used for solving four challenging constrained engineering design problems. Experimental results support the superiority of EDGNDO in solving the four problems. The superiority of EDGNDO in solving complex optimization problems is proven. The source code can be loaded from https://github.com/jsuzyy/EDGNDO.

Appendices

Appendix A

See Tables

Table 20 The ranking results of seven algorithms based on MAX and MEAN on CEC 2017 test suite with 30-dimensional

20,

Table 21 The ranking results of seven algorithms based on MIN and STD on CEC 2017 test suite with 30-dimensional

21,

Table 22 The ranking results of seven algorithms based on MAX and MEAN on CEC 2017 test suite with 50-dimensional

22,

Table 23 The ranking results of seven algorithms based on MIN and STD on CEC 2017 test suite with 50-dimensional

23

Appendix B

2.1 Mathematical expression of pressure vessel design problem

$$\begin{gathered} {\text{Minimize }}f\left( {T_{s} ,T_{h} ,R,L} \right) = f\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{3}^{2} + 3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3} \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = - x_{1} + 0.0193x_{3} \le 0 \hfill \\ g_{2} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = - x_{2} + 0.00954x_{3} \le 0 \hfill \\ g_{3} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = - \pi x_{3}^{2} x_{4} - \frac{4}{3}\pi x_{3}^{3} + 1296,000 \le 0 \hfill \\ g_{4} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = x_{4} - 240 \le 0 \hfill \\ \end{gathered}$$

where $\, 0 \le x_{i} \le 100, \, i = 1,2; \, 10 \le x_{i} \le 200, \, i = 3,4$.

2.2 Mathematical expression of welded beam design problem

$$\begin{gathered} {\text{Minimize }}f\left( {t,h,b,l} \right) = f\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = 1.10471x_{1}^{2} x_{2} + 0.04811x_{3} x_{4} \left( {14 + x_{2} } \right) \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = \tau \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) - \tau_{\max } \le 0 \hfill \\ g_{2} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = \sigma \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) - \sigma_{\max } \le 0 \hfill \\ g_{3} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = x_{1} - x_{4} \le 0 \hfill \\ g_{4} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = 0.10471x_{1}^{2} + 0.04811x_{3} x_{4} \left( {14 + x_{2} } \right) - 5 \le 0 \hfill \\ g_{5} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = 0.125 - x_{1} \le 0 \hfill \\ g_{6} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = \delta \left( x \right) - \delta_{\max } \le 0 \hfill \\ g_{7} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right) = P - P_{c} \left( x \right) \le 0 \hfill \\ 0.1 \le x_{i} \le 2 \, \, i{ = 1,4} \hfill \\ 0.1 \le x_{i} \le 10 \, i{ = 2,3} \hfill \\ \end{gathered}$$

$$\begin{gathered} {\text{where }}\left( x \right) = \sqrt {\left( {\tau^{^{\prime}} } \right)^{2} + 2\tau^{^{\prime}} \tau^{^{\prime\prime}} \frac{{x_{2} }}{2R} + \left( {\tau^{^{\prime\prime}} } \right)^{2} } ,\tau^{^{\prime}} = \frac{P}{{\sqrt 2 x_{1} x_{2} }},\tau^{^{\prime\prime}} = \frac{MR}{J},M = P\left( {L + \frac{{x_{2} }}{2}} \right),R = \sqrt {\left( {\frac{{x_{2} }}{2}} \right)^{2} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } , \hfill \\ J = 2\left( {\sqrt 2 x_{1} x_{2} \left( {\frac{{x_{2}^{2} }}{12} + \left( {\frac{{x_{1} + x_{3} }}{2}} \right)^{2} } \right)} \right),\sigma \left( x \right) = \frac{6PL}{{x_{4} x_{3}^{2} }},\delta \left( x \right) = \frac{{4PL^{3} }}{{Ex_{3}^{3} x_{4} }},P_{c} \left( x \right) = \frac{{4.013E\sqrt {\frac{{x_{3}^{2} x_{4}^{6} }}{36}} }}{{L^{2} }}\left( {1 - \frac{{x_{3} }}{2L}\sqrt{\frac{E}{4G}} } \right) \hfill \\ P = 6000{\text{lb}},L = 14{\text{in}},E = 30 \times 10^{6} {\text{psi}},G = 12 \times 10^{6} {\text{psi}},\tau_{\max } = 13,600{\text{psi}}, \, \sigma_{\max } = 30,000{\text{psi}},\delta_{\max } = 0.25{\text{in}} \hfill \\ \end{gathered}$$

2.3 Mathematical expression of speed reducer design problem

$$\begin{gathered} {\text{Minimize }}f\left( {b,m,z,l_{1} ,l_{2} ,d_{1} ,d_{2} } \right) = f\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = 0.7854x_{1} x_{2}^{2} \left( {3.3333x_{3}^{2} { + }14.9334x_{3} - 43.0934} \right) \hfill \\ - 1.508x_{1} \left( {x_{6}^{2} + x_{7}^{2} } \right) + 7.4777\left( {x_{6}^{3} + x_{7}^{3} } \right) + 0.7854\left( {x_{4} x_{6}^{2} + x_{5} x_{7}^{2} } \right) \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0 \hfill \\ g_{2} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{397.5}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0 \hfill \\ g_{3} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{1.93x_{4}^{3} }}{{x_{2} x_{6}^{4} x_{3} }} - 1 \le 0 \hfill \\ g_{4} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{1.93x_{5}^{3} }}{{x_{2} x_{7}^{4} x_{3} }} - 1 \le 0 \hfill \\ g_{5} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{\left( {\left( {\frac{{745x_{4} }}{{x_{2} x_{3} }}} \right)^{2} + 16.9 \times 10^{6} } \right)^{1/2} }}{{110x_{6}^{3} }} - 1 \le 0 \hfill \\ g_{6} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{\left( {\left( {\frac{{745x_{5} }}{{x_{2} x_{3} }}} \right)^{2} + 157.5 \times 10^{6} } \right)^{1/2} }}{{85x_{7}^{3} }} - 1 \le 0 \hfill \\ g_{7} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{x_{2} x_{3} }}{40} - 1 \le 0 \hfill \\ g_{8} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{5x_{2} }}{{x_{1} }} - 1 \le 0 \hfill \\ g_{9} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{x_{1} }}{{12x_{2} }} - 1 \le 0 \hfill \\ g_{10} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{1.5x_{6} + 1.9}}{{x_{4} }} - 1 \le 0 \hfill \\ g_{11} \left( {x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} } \right) = \frac{{1.1x_{7} + 1.9}}{{x_{5} }} - 1 \le 0 \hfill \\ \end{gathered}$$

where $2.6 \le x_{1} \le 3.6, \, 0.7 \le x_{2} \le 0.8,17 \le x_{3} \le 28,{ 7}{\text{.3}} \le x_{4} \le 8.3,{7}{\text{.3}} \le x_{5} \le 8.3,{ 2}{\text{.9}} \le x_{6} \le 3.9,{ 5}{\text{.0}} \le x_{7} \le 5.5$.

2.4 Mathematical expression of tension/compression design problem

$$\begin{gathered} {\text{Minimize }}f\left( {x_{1} ,x_{2} ,x_{3} } \right) = \left( {x_{3} + 2} \right)x_{2} x_{1}^{2} \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {x_{1} ,x_{2} ,x_{3} } \right) = 1 - \frac{{x_{2}^{3} x_{3} }}{{71,785x_{1}^{4} }} \le 0 \hfill \\ g_{2} \left( {x_{1} ,x_{2} ,x_{3} } \right) = 4x_{2}^{2} - \frac{{x_{1} x_{2} }}{{12.566\left( {x_{2} x_{1}^{3} - x_{1}^{4} } \right)}} + \frac{1}{{5108x_{1}^{2} }} - 1 \le 0 \hfill \\ g_{3} \left( {x_{1} ,x_{2} ,x_{3} } \right) = 1 - \frac{{140.45x_{1} }}{{x_{2}^{2} x_{3} }} \le 0 \hfill \\ g_{4} \left( {x_{1} ,x_{2} ,x_{3} } \right) = x_{2} + \frac{{x_{1} }}{1.5} - 1 \le 0 \hfill \\ \end{gathered}$$

where $\, 0.05 \le x_{1} \le 2, \, 0.25 \le x_{2} \le 1.30, \, 2.00 \le x_{3} \le 15.00$.